Advanced geometry of skew and straight bevel gears produced by forging

ABSTRACT

The present invention provides an advanced geometry for skew and straight bevel gears having a localized and more stable bearing contact produced by forging. The localization of bearing contact is achieved by substitution of flat tooth surfaces with parabolic tooth surfaces. This is accomplished by either modifying line-contact directly or by using a generating parabolic cylinder during generation of the tooth surfaces. The actual contact area is spread over an ellipse centered around an instantaneous theoretical point of contact at the apex of the parabolic tooth surface. This localized bearing contact reduces the shift of the bearing due to assembly and manufacturing errors, thereby providing a more durable and quieter gear. Furthermore, the geometry of the gear enables creation of dies from which the gear may be forged. Transmission errors are modeled using a pre-designed parabolic function that coincides with the parabolic tooth surface design of the present invention.

FIELD OF THE INVENTION

[0001] The present invention relates to the design of the geometry for skew and straight bevel gears produced by forging.

BACKGROUND OF THE INVENTION

[0002] In the past, skew and straight bevel gears have been designed to have line-contact between a gear and pinion. Here, the individual teeth of the gear and pinion interact with one another along lines of contact that shift along the face of the teeth as the gear and pinion rotate. However, errors in assembly or manufacturing, such as misalignment of the gear and pinion, can lead to the area of contact being transferred to the edges of the teeth. A transmission function φ₂(φ₁) of misaligned prior art gears is illustrated in FIG. 1. Angles φ₁ and φ₂ of transmission function φ₂(φ₁) represent angles of rotation of a prior art gear and pinion. Function φ₂(φ₁) exists as a sum of a linear function and a discontinuous almost linear function of transmission errors caused by errors of alignment. See “Gear and Applied Theory,” by F. L. Litvin, Prentice Hall (1994). Such transmission errors cause large acceleration when one pair of teeth is changed for another, resulting in inevitable vibration and noise of the gear drives.

[0003] Moreover, many of the existing skew and bevel gears are designed to be produced by cutting. The manufacture of gears produced by cutting is undesirable for at least two reasons. First, cutting the tooth surface does not guarantee reduction of noise and good bearing contact. Second, due to wear on the cutting tool, consistency between gear sets is difficult to obtain. As an alternative to cutting, forging of gears is preferred.

[0004] Forging is preferred over cutting because it allows the optimal geometry of gears to be chosen so as to improve bearing contact and reduce transmission errors. The optimal geometry can be easily obtained through the use of the proper dies. Application of the proper die geometry provides a localized bearing contact and a parabolic function of transmission errors. Such a parabolic function of transmission errors is able to absorb the linear function of transmission errors caused by misalignment and avoids the noise and vibration caused by misalignment. See “Gear and Applied Theory,” by F. L. Litvin, Prentice Hall (1994).

SUMMARY OF THE INVENTION

[0005] The present invention provides a geometry for skew and straight bevel gears produced by forging having a localized and stable bearing contact between tooth surfaces and a function of transmission errors having a favorable shape. The localized bearing contact of the pinion-gear tooth surface is located in the central part of the tooth surfaces. Localization of the bearing contact may be achieved in two ways.

[0006] Localization of the bearing contact may be achieved through modification of the lines of contact of one of the mating gears, either the pinion or the gear, so that they correspond to the meshing of the mating gear with the crown gear. Modification of the lines of contact is achieved by the parabolic deviation of the theoretical line of contact so that the localized bearing contact is orientated across the tooth surface and located in the central part of the tooth surface.

[0007] In the second approach, localization of the bearing contact is achieved using two generating surfaces of an imaginary crown gear to separately generate the pinion and gear tooth surfaces. The generating surfaces represent a plane and a parabolic cylinder. The actual contact area between the teeth is spread over an ellipse centered around an instantaneous theoretical point of contact. This localized bearing contact reduces the shifting of the bearing area to the edges of the tooth surface due to assembly and manufacturing errors, such as misalignment, thereby producing a more durable and more quiet gear. Further, the geometry of the gear is sufficiently simple such that dies used for forging may be easily produced.

[0008] The sensitivity of bevel gears with the proposed geometry is reduced due to the use of a pre-designed parabolic function of transmission errors. The function is able to absorb the almost linear functions of transmission errors caused by misalignment. See “Gear and Applied Theory,” by F. L. Litvin, Prentice Hall (1994). The transmission function of the gear drive is provided by application of a nonlinear function that relates the angles of rotation of the bevel gear to the crown gear wherein meshing of the gear is performed. The transmission function is an algebraic sum of linear and parabolic functions.

[0009] Producing skew and straight bevel gears by forging using Applicants' geometry will not result in increased production costs when compared to production costs associated with using the existing geometry. The cost will not be affected because forging is based on the application of inexpensive dies and because the die surfaces are generated point by point. Therefore, the generation of dies of the new geometry does not present additional production difficulties.

BRIEF DESCRIPTION OF THE DRAWINGS

[0010] The present invention will become more fully understood from the detailed description and the accompanying drawings, wherein:

[0011]FIG. 1 is a graphical illustration of transmission function φ₂(φ₁) produced by the prior art gear geometry.

[0012]FIG. 2(a) is a graphical illustration of the generating space and planes of an imaginary crown gear generated in coordinate system S_(b)(x_(b),y_(b),z_(b)) in accordance with the present invention;

[0013]FIG. 2(b) is a graphical illustration of the formation of the generating space and planes of an imaginary crown gear generated in coordinate system S_(b)(x_(b),y_(b),z_(b)) in accordance with the present invention;

[0014]FIG. 3 shows the installation of coordinate system S_(c)(x_(c),y_(c),z_(c)) with respect to the coordinate system S_(b)(x_(b),y_(b),z_(b)) of an imaginary crown gear in accordance with the present invention.

[0015]FIG. 4 shows the rotation of coordinate systems S_(b)(x_(b),y_(b),z_(b)) and S_(c)(x_(c),y_(c),z_(c)) of an imaginary crown gear with respect to a fixed coordinate system S_(h)(x_(h),y_(h),z_(h)) in accordance with the present invention;

[0016]FIG. 5 shows the relative rotations of a pinion with coordinate system S₁(x₁,y₁,z₁) and a gear with coordinate system S₂(x₂,y₂,z₂) with respect to fixed coordinate system S_(h)(x_(h),y_(h),z_(h)) in accordance with the present invention;

[0017]FIG. 6 shows a schematic of left-hand skew teeth in accordance with the present invention;

[0018]FIG. 7 shows a schematic of right-hand skew teeth in accordance with the present invention;

[0019]FIG. 8 shows the modification of theoretical lines of contact on a tooth surface in accordance with the present invention;

[0020]FIG. 9 shows the area of contact and path of contact on a tooth surface in accordance with the present invention; and

[0021]FIG. 10 shows a generating plane and a generating cylinder in accordance with the present invention.

[0022]FIG. 11 is a graphical illustration of a transmission function φ₂(φ₁), the transmission function is the sum of a linear function and a parabolic function of transmission errors predesigned for absorption of linear functions of transmission errors caused by misalignment of gears in accordance with the present invention.

[0023]FIG. 12 shows a crown gear fillet in accordance with the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0024] The following description of the preferred embodiment is merely exemplary in nature and is in no way intended to limit the invention, its application, or uses.

[0025] The mathematical generation of gear and pinion tooth surfaces is achieved using an imaginary crown gear that is in mesh with the pinion and gear tooth surfaces. The geometry of the pinion and gear tooth surfaces is generated as envelopes to a generating plane on the imaginary crown gear. This mathematical generation of the tooth surfaces is preferably done with a computer simulation using the method described herein.

[0026] Referring now to the drawing figures, the imaginary crown gear is shown in FIG. 2(a). The imaginary crown gear is formed by two generating planes that are conjugated to the respective tooth sides of the pinion and the gear. The generating planes Σ_(b) and Σ_(b)* are represented in auxiliary coordinate system S_(b) defined by axes X_(b)-Y_(b)-Z_(b). Each of the generating planes are formed by two straight lines. For example, generating plane Σ_(b) is formed by lines O_(b)D and O_(b)E, as shown in FIG. 2(b). Parameter 2μ represents the width of the space in the pitch plane, parameter α is the pressure angle, and auxiliary parameter q is determined by the formula

tan q=sin μ cot α  (1)

[0027] To generate the pinion and gear tooth surfaces, coordinate system S_(b)(x_(b),y_(b),z_(b)) of the crown gear is rigidly connected to coordinate system S_(c)(x_(c),y_(c),z_(c)). Coordinate system S_(b) is installed with respect to S_(c) as shown in FIG. 3, wherein a skew bevel gear is considered. In the case where a straight bevel gear is generated, parameters α and ρ are equal to zero and coordinate system S_(b) coincides with S_(c).

[0028] During the generation of pinion and gear tooth surfaces, the crown gear (with coordinate systems S_(c) and S_(b)) rotates about axis X_(h) of fixed coordinate system S_(h) defined by axes X_(h)-Y_(h)-Z_(h) as shown in FIG. 4. Angle ψ_(c) represents the instantaneous angle of rotation of S_(c) and of S_(b) with respect to S_(h) performed during generation. During rotation of the crown gear, axis Z_(b) remains tangent to the circle of radius ρ that is traced out by point Q of axis Z_(b), as shown in FIGS. 3 and 4.

[0029] Simultaneously with rotation of the imaginary crown gear, the pinion and the gear perform related rotations about axes Z₁ and Z₂, as seen in FIG. 5. The pinion is represented in coordinate system S₁ defined by axes X₁-Y₁-Z₁, and the gear is represented in coordinate system S₂ defined by axes X₂-Y₂-Z₂. Arrows ψ_(c), ψ₁, and ψ₂ in FIG. 5 illustrate rotations of the imaginary crown gear, pinion, and gear, respectively. Angles γ₁ and γ₂ represent angles of pitch cones of the pinion and gear.

[0030] During the process of the related rotations discussed above, the generating plane Σ_(b) of the imaginary crown gear, shown in FIG. 2, generates a family of planes in coordinate systems S₁ and S₂, shown in FIG. 5, that are rigidly connected to the pinion and the gear. The tooth surfaces of the pinion and the gear are determined as the envelopes to the family of Σ_(b) planes. See “Gear Geometry and Applied Theory,” by F. L. Litvin, Prentice Hall, 1994. However, the envelopes are modified as will be explained below.

[0031] The installment of S_(b) and S_(c) with respect to S_(h) determines the direction of the skew teeth. FIG. 6 illustrates teeth skewed in a left-hand direction while FIG. 7 illustrates teeth skewed in a right-hand direction. Linear skew teeth are designated by reference number 10 and illustrated with a solid line. Parabolic skew teeth generated by the method described by the present invention are designated by reference numeral 20 and are illustrated with a dashed line. The mean skew angle β is determined as:

sin β=ρ/R _(e)  (2)

[0032] wherein R_(e) is the outer radius of the crown gear.

[0033] During the imaginary pinion and gear generation process, the pinion and gear are in line-contact with the imaginary crown gear. To achieve localization of the bearing contact, the line-contact along the tooth surfaces must be substituted by point contact. Two alternative methods for achieving point contact between the pinion and gear tooth surfaces are proposed.

[0034] In the first embodiment, shown in FIG. 8, the theoretical lines of contact on the tooth surface of the pinion and the gear are modified after generation of the tooth surfaces using the method described above. The family of theoretical lines of contact on the pinion or gear tooth surface is designated by L₁. The modified line is designated as L₁*. The deviation of L₁* from L₁ satisfies the following conditions: the magnitude of deviation is determined by a parabolic function wherein the deviation at point M is equal to zero, and it is accomplished along normal N to the tooth surface. The deviation described above creates a path of contact on the mating tooth surfaces consisting of a locus of points M, located in the center of the tooth surfaces, wherein the deviation is equal to zero. The actual contact is spread at each point over an elliptical area centered around the locus of points M due to elastic deformation of the mating tooth surfaces, as shown in FIG. 9.

[0035] In the second embodiment, localization of bearing contact is achieved using two crown gear generating surfaces that generate pinion and gear tooth surfaces separately. One of the generating surfaces is a plane and the other generating surface is a parabolic cylinder, as shown in FIG. 10. The two generating surfaces replace the generating surfaces Σ_(b) and Σ*_(b) in FIGS. 2(a) and 2(b). The line of tangency of the parabolic cylinder and the plane is the path of contact on the generating surface. The gear and pinion are designed using the method described above wherein the generating parabolic cylinder generates, as it rotates, a parabolic tooth surface that localizes the bearing contact when the gear and pinion are meshed.

[0036] This advanced geometry enables the use of a pre-designed parabolic function that absorbs linear functions of transmission errors caused by misalignment. This transmission function φ₂(φ₁) is represented as the sum of a linear function and a parabolic function of transmission errors as graphically illustrated in FIG. 11. The transmission function φ₂(φ₁) is designated by angles of rotation φ₁ and φ₂ generated during the process of meshing of the pinion and gear in the gear drive. However, angles of rotation φ₁ and φ₂ are different from angles of rotation ψ_(c), ψ₁, and ψ₂ generated during the process of gear generation seen in FIG. 4 and FIG. 5.

[0037] During the generation process, the following relation is provided between angles of rotation ψ_(c) of the crown gear and ψ₂ of the gear where k is the parabolic coefficient:

ψ₂(ψ_(c))=(N _(c) /N ₂)ψ_(c) −k(ψ_(c))².  (3)

[0038] The angles of rotation of the pinion and the crown gear in the gear and pinion generation process are related according to the following equation:

ψ₁/ψ_(c) =N _(c) sin γ₁ /N ₁.  (4)

[0039] The computerized simulation used by Applicants to generate the pinion and gear results in the following gear drive transmission function:

φ₂(φ₁)=(N ₁ /N ₂)φ₁ −k(sin γ₁)²(φ₁)².  (5)

[0040] The above gear drive transmission function provides a parabolic function of transmission errors as follows:

Δφ₂(φ₁)=−k(sin γ₁)²(φ₁)².  (6)

[0041] Application of the predesigned parabolic function of transmission errors allows for the absorption of linear functions of transmission errors caused by misalignment.

[0042] The surfaces of the pinion and gear are determined numerically as envelopes of the generating surfaces using approaches represented in differential geometry as well as in gear theory. See “Gear and Applied Theory,” by F. L. Litvin, Prentice Hall (1994). For instance, the pinion tooth surface Σ₁ is determined as the envelope to the crown gear Σ_(c) by the following equations:

r ₁(u _(c),θ_(c),ψ_(c))=M _(1c)(ψ_(c))r _(c)(u _(c),θ_(c))  (7) $\begin{matrix} {{\left( {\frac{\partial r_{1}}{\partial u_{c}} \times \frac{\partial r_{1}}{\partial\theta_{c}}} \right) \cdot \frac{\partial r_{1}}{\partial\psi_{c}}} = {{f_{c1}\left( {u_{c},\theta_{c},\psi_{c}} \right)} = 0.}} & (8) \end{matrix}$

[0043] Here r_(c)(u_(c),θ_(c)) is the vector equation that represents the generating surface Σ_(c) of the crown gear; (u_(c),θ_(c)) are the surface parameters of Σ_(c); matrix M_(1c) describes the coordinate transformation from coordinate system S_(c)(x_(c),y_(c),z_(c)) rigidly connected to the crown gear to coordinate system S₁(x₁,y₁,z₁) rigidly connected to the pinion; ψ_(c) is the generalized parameter of motion that defines angles of rotation of the crown gear and the pinion; vector function r₁(u_(c),θ_(c),ψ_(c)) defines the family of generating surfaces Σ_(c) in coordinate system S₁. Above equation (8) is the equation of meshing f_(c1)=0. Above equations (7) and (8) determine pinion tooth surface Σ₁ by three related parameters. The tooth surfaces Σ₂ of the gear are determined similarly.

[0044] The pinion and gear tooth surfaces Σ₁ and Σ₂ formed as discussed above are made as exact copies of respective dies. The dies are generated point by point using a Computer Numerically Controlled (CNC) machine. The pinion and gear with the desired surface characteristics are made from dies which are an exact copy of their respective surfaces. A metal or other suitable material is then placed into the die and allowed to cool, forging a bevel gear with the proposed geometry of the present invention. The result is a pinion and gear with teeth having curved surfaces. The curved surfaces of the pinion teeth are tangent to the curved surfaces of the gear teeth such that interaction between teeth occurs substantially at a common point of each curved mating tooth surface.

[0045] The gear and pinion may also contain a fillet. The fillet of the gear or pinion is generated by the fillet of the crown gear which is illustrated in FIG. 12(a). The crown gear fillet is formed by two cones and a plane (FIG. 2(b)). However, in some cases the crown gear fillet may be formed by one cone. The fillet of the pinion (or the gear) is generated as the envelope to the fillet of the generating crowning gear.

[0046] The description of the invention is merely exemplary in nature and, thus, variations that do not depart from the gist of the invention are intended to be within the scope of the invention. Such variations are not to be regarded as a departure from the spirit and scope of the invention. 

What is claimed is:
 1. A gearing system produced by forging, the gearing system comprising: a gear having a plurality of teeth; and a pinion having a plurality of teeth; wherein the gear is in meshing contact with the pinion such that the gear teeth and pinion teeth are in substantially point contact, the meshing contact accompanied by a parabolic function of transmission errors.
 2. The gearing system of claim 1, wherein the pinion and gear are of a straight bevel type.
 3. The gearing system of claim 1, wherein the pinion and gear are of a skew type.
 4. The gearing system of claim 1, wherein the contact between the gear teeth and the pinion teeth occurs substantially within the center of the gear teeth and the pinion teeth.
 5. The gearing system of claim 1, wherein the instantaneous contact area between the pinion teeth and the gear teeth has a substantially elliptical shape.
 6. The gearing system of claim 5, wherein the instantaneous elliptically shaped contact area has a center point, the center point being a theoretical point of tangency between the pinion teeth and the gear teeth.
 7. The gearing system of claim 1, wherein the pinion teeth and the gear teeth have double curvature surfaces, the curvature surfaces of the pinion teeth and the gear teeth having a common tangent plane at the points of meshing contact between the gear and the pinion.
 8. The gearing system of claim 1, wherein the contact between the pinion teeth and gear teeth is localized as a result of modification of lines of contact of either the pinion, the gear or both the pinion and the gear, wherein the lines of contact are obtained by imaginary meshing of the pinion or gear or both with a crown gear, the tooth surfaces of the crown gear being planes wherein the modification of lines of contact is performed as a deviation from the theoretical line of contact along the normal to the surface and the magnitude of deviation is a parabolic function.
 9. The gearing system of claim 1, wherein the contact between the pinion teeth and the gear teeth is localized through the separate generation of the tooth surfaces of the pinion and gear by application of two crown gears where the tooth surface of one of the crown gears is a plane and the tooth surface of the other crown gear is a parabolic cylinder, wherein the crown gears, the pinion, and the gear perform related motions.
 10. The gearing system of claim 1, wherein the parabolic transmission error function is calculated using the algebraic sum of a linear function and a parabolic function that relates the angles of rotation of the gear and pinion with an imaginary crown gear.
 11. The gearing system of claim 1, wherein the pinion and gear are made from dies having a surface which is an inverted copy of the pinion or gear tooth surface.
 12. A method for designing the geometry of a bevel gear system, the method comprising the steps of: providing an imaginary crown gear defined by a generating plane and a generating parabolic cylinder; providing a gear; providing a pinion; meshing said imaginary crown gear with said gear and with said pinion; rotating said imaginary crown gear about a first axis; rotating said gear about a second axis; rotating said pinion about a third axis; and generating, as envelopes to said generating parabolic cylinder, parabolic tooth surfaces on said pinion and said gear as said imaginary crown gear, said gear, and said pinion rotate.
 13. The method of designing the geometry of a bevel gear system of claim 12, further comprising the step of preparing a die from said generated tooth surfaces.
 14. The method of designing the geometry of a bevel gear system of claim 13, further comprising the step of forging a gear or pinion from said die, wherein the surface of said die is an inverted copy of said generated tooth surfaces.
 15. The method of designing the geometry of a bevel gear system of claim 12, further comprising the step of preparing a pre-designed parabolic transmission error function.
 16. The method of designing the geometry of a bevel gear system of claim 15, wherein said pre-designed parabolic transmission error function is calculated using the algebraic sum of a linear function and a parabolic function that relates the angles of rotation of said gear and said pinion with said imaginary crown gear.
 17. A method for designing the geometry of a bevel gear system, the method comprising the steps of: providing an imaginary crown gear defined by two generating planes; providing a gear; providing a pinion; meshing said imaginary crown gear with said gear and with said pinion such that said generating planes are conjugated to the tooth sides of said pinion and said gear providing line contact between said imaginary crown gear and said pinion and said gear; rotating said imaginary crown gear about a first axis; rotating said gear about a second axis; rotating said pinion about a third axis; generating, as envelopes to said generating planes, tooth surfaces on said pinion and said gear as said imaginary crown, said gear, and said pinion rotate; modifying said line contact between said imaginary crown gear and said gear by deviating from said line contact by a parabolic function normal to said tooth surfaces on said gear; and modifying said line contact between said imaginary crown gear and said pinion by deviating from said line contact by a parabolic function normal to said tooth surfaces on said pinion.
 18. The method of designing the geometry of a bevel gear system of claim 17, further comprising the step of preparing a die from said generated tooth surfaces.
 19. The method of designing the geometry of a bevel gear system of claim 18, further comprising the step of forging a gear or pinion from said die, wherein the surface of said die is an inverted copy of said generated tooth surfaces.
 20. The method of designing the geometry of a bevel gear system of claim 17, further comprising the step of preparing a pre-designed parabolic transmission error function.
 21. The method of designing the geometry of a bevel gear system of claim 20, wherein said pre-designed parabolic transmission error function is calculated using the algebraic sum of a linear function and a parabolic function that relates the angles of rotation of said gear and said pinion with said imaginary crown gear. 